On complete convergence in mean for double sums of independent random elements in Banach spaces

Abstract

For a double array of random elements {T_m,n, m ≥ 1, n ≥ 1} in a real separable Banach space X, we study the notion of T_m,n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T_m,n converging completely to 0 and (ii) T_m,n converging to 0 in mean of order p as max{m, n} →∞. When X is of Rademacher type p (1 ≤ p ≤ 2), for a double array of independent mean 0 random elements {V_m,n, m ≥ 1, n ≥ 1} in X and a double array of constants {b_m,n, m ≥ 1, n ≥ 1}, conditions are provided under which max_{1≤k≤m,1≤l≤n}||Ʃ_i=1^kƩ_j=1^lV_i,j||/b_m,n converges completely to 0 in mean of order p. Moreover, these conditions are shown to provide an exact characterization of Rademacher type p (1 ≤ p ≤ 2) Banach spaces. Illustrative examples are provided.